Do you mean a particle is restricted to move on a torus surface (inside a torus and always in contact with the surface)?

What is the interaction force between particle and the torus surface? Normal force ?
or Is there another force to keep particle always in contact with the torus surface (it might fall down due to gravity).

Assume the larger radius is R, and smaller radius is r for the torus.
Does z=r \sin\beta, and \tan\gamma=y/x in your case?
I do not understand the meaning of \alpha in the attached picture.
Your problem might be similar to a circular motion in vertical direction , plus a constant angular speed rotational motion (angular momentum is conserved).